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(1)1) Consider the following formula, where P,Q,R are propositional variables

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(1)

1)

Consider the following formula, where P,Q,R are propositional variables:

[ ( P  Q )   R ]  [ ( R  P )   Q ]

Rewrite it in the CNF form (1 point) and answer to the following questions:

Is the answer to the SAT problem for this formula YES or NO (1 point)?

(2)

2)

Consider the expert system based on the following rules:

IF the engine is getting gas AND the engine turns over

THEN the problem is spark plugs IF the engine does not turn over AND the lights do not come on

THEN the problem is battery or cables IF the engine does not turn over AND the lights come on

THEN the problem is the starter motor IF there is gas in the fuel tank AND there is gas in the carburator

THEN the engine is getting gas

Rewrite these rules as clauses (2 points), using the following interpretation of propositional variables:

F1 : the engine is getting gas F2 : the engine turns over F3 : the lights come on

F4 : there is gas in the fuel tank F5 : there is gas in the carburator P1 : the problem is spark plugs P2 : the problem is battery P3 : the problem is cables

P4 : the problem is the starter motor

Construct the resolution tree showing that if there is gas in the fuel tank and the carburator, and we know that there are no problems with battery, cables, as well as the starter motor, then there must be the problem with spark plugs (2 points).

(3)

3)

Consider the following optimization problem concerning undirected graphs G=(V,E), where V denotes the set of nodes and EVV denoted the set of undirected edges linking the nodes.

Problem: Given G=(V,E), find the minimal dominating set, that is the smallest subset XV such that any node xX is linked by an edge eE with some yX.

Design the greedy algorithm for solving the above problem (1 point).

Explain (step by step) the performance of your algorithm (1 point) using the following example of the graph G=(V,E) :

where

V = {v1,v2,v3,v4,v5,v6,v7,v8,v9}

and

E = {(v1,v4),(v1,v5),(v2,v3),(v2,v5),(v2,v6),(v3,v6), (v4,v5),(v4,v7),(v6,v7),(v6,v9),(v7,v8),(v8,v9)}

Explain how the SAT problem can be reduced to the minimal dominating set problem (3 points).

v1 v2

v4

v7 v8 v9

v6 v5

v3

Cytaty

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